Effect of electric fields on the electronic and thermoelectric properties of zigzag buckling silicene nanoribbons

In this paper, we examine the zigzag edge of buckling silicene nanoribbons (BSiNRs) with the width W, which can also be characterised by M zigzag lines in which $W=\left(3M-2\right){a}_{0}/2$ where ${a}_{0}=0.224\,{\rm{nm}}$ is the distance between the two nearest Si-Si atoms, as shown in figure 1(a). BSiNRs contain the sp²-hybridised Si-atoms (yellow spheres) in the lower layer and the sp³-hybridised Si atoms (blue spheres) in the upper layer. The distance between these two layers is ${\rm{\Delta }}d=0.044\,{\rm{nm}}$ (as indicated in figure 1(b)). The hopping energy between the two nearest neighbouring atoms is presented by the parameter t.

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To investigate the effect of external electric fields, electric gates are arranged as schematised in figure 1(b). We assumed that the vertical electric field was induced by two gates as ${V}_{t}/2$ and $-{V}_{t}/2$ (as indicated by orange colour), which are set up above and below the studied system, respectively. Besides, two side gates with the voltages as ${V}_{t}/2$ and ${V}_{s}/2$ (as shown in blue colour) are designed to generate a transverse electric field along the y-axis.

To examine the electronic properties of zigzag BSiNRs, we used the tight-binding (TB) method. The TB Hamiltonian is presented as follows [34]:

$$\begin{eqnarray}&&H=\displaystyle \sum _{i}{\varepsilon }_{i}\left|i\right\rangle \left\langle i\right|+\displaystyle \sum _{\left\langle ij\right\rangle }{t}_{ij}\left(\left|i\right\rangle \left\langle j\right|+\left|j\right\rangle \left\langle i\right|\right),\end{eqnarray} \tag{ 1 }$$

in which, ${\varepsilon }_{i}$ is the onsite energy of Si atom at the i-th site, ${t}_{ij}$ is the energy coupling between two Si atoms at the i-th and j-th sites. It is worth mentioning that the spin-orbit coupling was not taken into account in the model (1) for the sake of simplicity and avoiding a computational burden. This model has also been considered to be appropriate for the low-energy states [34] which are the most relevant for the transport of electrons.

In our calculations, only the nearest neighbouring hopping t is considered as in [34]. The value $t=-1.6\,{\rm{eV}}$ was taken from [19, 35]. For the flat ribbon structure of silicene, ${\varepsilon }_{i}=0.$ However, for a buckling system, an energy factor H_B (named as the buckling energy) was added to the Hamiltonian (1). Furthermore, to take into account the presence of external electric fields in calculations, beyond the components of the Hamiltonian denoted above, a matrix U_ext presenting the electrostatic potential energy induced by the external electric fields is also added and the full Hamiltonian can be rewritten as:

$$\begin{eqnarray}&&{H}_{F}=H+{H}_{B}+{U}_{ext}.\end{eqnarray} \tag{ 2 }$$

To be convenient in the deployment of components in equation (2), the studied zigzag BSiNRs were divided into unit cells (as illustrated in figure 1(a)) in which each unit cell contains 2 M atoms. The matrices of the Hamiltonian of a certain cell 0 and the couplings with its neighbouring cells −1 and 1 were calculated by

$$\begin{eqnarray*}\begin{array}{c}{H}_{00}=\left[\begin{array}{ccccc}\begin{array}{cc}0 & t\\ t & \,0\,\end{array} & \begin{array}{cc}0 & 0\\ t & 0\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \mathrm{...}\\ \begin{array}{cc}0 & t\\ 0 & 0\end{array} & \begin{array}{cc}0 & t\\ t & \,0\,\end{array} & \begin{array}{cc}0 & 0\\ t & 0\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \mathrm{...}\\ \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}0 & t\\ 0 & 0\end{array} & \begin{array}{cc}\,0 & t\\ t & \,0\,\end{array} & \begin{array}{cc}0 & 0\\ t & 0\end{array} & \mathrm{...}\\ \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}0 & t\\ 0 & 0\end{array} & \begin{array}{cc}\,0 & t\\ t & 0\,\end{array} & \mathrm{...}\\ \vdots & \vdots & \vdots & \vdots & \ddots \end{array}\right];\\ {H}_{0-1}=\left[\begin{array}{ccccc}\begin{array}{cc}0 & t\\ 0 & \,0\,\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \mathrm{...}\\ \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}0 & 0\\ t & \,0\,\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \mathrm{...}\\ \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}0 & t\\ 0 & \,0\,\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \mathrm{...}\\ \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}0 & 0\\ t & 0\,\end{array} & \mathrm{...}\\ \vdots & \vdots & \vdots & \vdots & \ddots \end{array}\right];\\ {H}_{01}=\left[\begin{array}{ccccc}\begin{array}{cc}0 & 0\\ t & \,0\,\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \mathrm{...}\\ \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}0 & t\\ 0 & \,0\,\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \mathrm{...}\\ \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}\,0 & 0\\ t & \,0\,\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \mathrm{...}\\ \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}0 & 0\\ 0 & 0\end{array} & \begin{array}{cc}0 & t\\ 0 & \,0\,\end{array} & \mathrm{...}\\ \vdots & \vdots & \vdots & \vdots & \ddots \end{array}\right].\end{array}\end{eqnarray*}$$

The buckling matrix H_B was constructed by the parameter Δ, with Δ = 3.9 meV [35] presenting the difference of the hybridisation of Si atoms. This matrix only has nonzero terms in the main diagonal line:

$$\begin{eqnarray}{H}_{B}={\left[\begin{array}{ccccc}-{\rm{\Delta }} & 0 & 0 & \mathrm{...} & 0\\ 0 & {\rm{\Delta }} & 0 & \mathrm{...} & 0\\ 0 & 0 & -{\rm{\Delta }} & \mathrm{...} & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \mathrm{...} & {\rm{\Delta }}\end{array}\right]}_{\left(2M\times 2M\right)}.\end{eqnarray} \tag{ 3 }$$

The U_ext was divided into two parts for the individual contribution of the vertical and transverse electric fields: ${U}_{ext}={U}_{\perp }+{U}_{\parallel }.$ Similar to the impact of the buckling parameter, the perpendicular electric field induces an opposite impact on atoms in the top and bottom layers (as seen in figure 1(b)) with the potentials equal to $-e{V}_{t}/2$ and $e{V}_{t}/2,$ respectively. On the other hand, the electrostatic potential induced by the transverse electric field depends on the position of atoms, from one edge to the other one of the considered Z-BSiNR, i.e., $U\left(i\right)=-eV\left(i\right)$ with $V\left(i\right)\,=-{V}_{s}/2+E\times d\left(i\right)$ and here $E={V}_{s}/W,$ $d\left(i\right)$ is the distance of the i-th atom along the ribbon width that is presented in table 1. The matrix form of ${U}_{\parallel }$ is written as

$$\begin{eqnarray}{U}_{\parallel }={\left[\begin{array}{ccccc}{U}_{1} & 0 & 0 & \mathrm{...} & 0\\ 0 & {U}_{2} & 0 & \mathrm{...} & 0\\ 0 & 0 & {U}_{3} & \mathrm{...} & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \mathrm{...} & {U}_{2M}\end{array}\right]}_{\left(2M\times 2M\right)}.\end{eqnarray} \tag{ 4 }$$

Table 1. Distance of atoms along the y-direction $\left({y}_{0}={a}_{0}/2\right).$

$i$	1	2	3	4	5	6	...	2M-1	2 M
$d(i)$	$0{y}_{0}$	$1{y}_{0}$	$3{y}_{0}$	$4{y}_{0}$	$6{y}_{0}$	$7{y}_{0}$	...	$\left(3M-3\right){y}_{0}$	$\left(3M-2\right){y}_{0}$

The energy bands were obtained from equation:

$$\begin{eqnarray}&&h{\phi }_{0}=E{\phi }_{0},\end{eqnarray} \tag{ 5 }$$

with $h={H}_{00}+{H}_{B}+{U}_{ext}+\displaystyle {\sum }_{i=-1,1}{H}_{0i}{e}^{i\overrightarrow{k}{\overrightarrow{R}}_{0i}}.$

In this work, quantum transport and the thermoelectric properties of zigzag BSiNRs were also investigated. The thermoelectric conversion is evaluated by the figure of merit $ZT$ which is defined by [36]:

$$\begin{eqnarray}&&ZT=\displaystyle \frac{P}{K}.T,\end{eqnarray} \tag{ 6 }$$

where $P={G}_{e}.{S}^{2}$ is the power factor, $K={K}_{e}+{K}_{p}$ is the total thermal conductance, T is the absolute temperature. And ${G}_{e},S,{K}_{e}$ and ${K}_{p}$ are electrical conductance, Seebeck coefficient, electron thermal conductance and phonon thermal conductance, respectively. The electronic part including ${G}_{e},\,S,\,{K}_{e}$ was calculated using non-equilibrium Green's function technique and the Landauer formalism, which is presented in detail in [37, 38]. The phonon conductance ${K}_{p}$ was referred from a previous study using molecular dynamics simulations [39]. It is also worth mentioning that other methodologies such as the Boltzmann transport equation can describe transport properties of both electrons and phonons. However, such a method actually considers the dynamics of particles classically [40] and lacks the capacity to treat quantum effects. On the other hand, the NEGF directly takes into account the presence of defects incorporated into the Hamiltonian and the Landauer formalism can treat the natural quantum effects of transport at the nanoscale. We therefore chose the NEGF and the Landauer formalism as relevant methods to study transport properties in this work.

First, we investigate the energy bands of zigzag BSiNRs in the presence of electric fields. To better understand the difference between the electronic structure of a zigzag BSiNR and that of its counterparts, we also examine the energy bands of zigzag flat silicene nanoribbons (FSiNRs) and zigzag graphene nanoribbons (GNRs) for comparison.

In figures 2(a)–(c), the band structures of zigzag nanoribbons of width M = 12 are shown for graphene, flat silicene, and buckling silicene, respectively. As can be observed, flat bands at the zero energy are present in all the considered structures. Such flat bands are well known as edge states that are strongly localised at the two edges of the zigzag ribbon [41–43]. An energy gap is not observed in these band structures, pointing out that the metallic feature is not unique for zigzag GNRs [43, 44], but it could be a general characteristic for all ribbons with zigzag edges.

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Interestingly, comparing the dispersive parts of the lowest conduction (or the highest valence) band of these electronic structures, it is visible that the band of the zigzag GNR has a slope that is greater than the ones of the zigzag silicene ribbons (both flat and buckling forms). Such a result indicates that the electron mobility in GNRs is higher than that in other silicene structures.

To look more closely at the energy distribution around the Fermi level (E = 0), in figure 2(d) we put the energy bands of all the three structures and the narrow range of energy between −0.05 eV and 0.05 eV is examined. As can be seen, a very small gap (about 7.8 meV ≈ $2{\rm{\Delta }}$) between two energy levels of the conduction and valence bands is observed in the BSiNR but not in the others, showing an interesting difference between buckling and flat layer structures. Such a result could be understood based on the hybridisation of atoms in the structure, i.e., BSiNRs consist of not only sp²-hybridised atoms (like GNRs and FSiNRs) but also sp³-hybridised atoms [12, 45, 46]. The existence of sp³ hybrid orbital weakens or leads to the disappearance of several non-local pi bonds on the surface of the material and creates asymmetry in the structure, reducing the electrical conduction of the material. FSiNRs have the same type of hybridisation (sp²) as that in GNRs, therefore they share similar electronic characteristics even though they have different bonding lengths (0.224 nm in silicene and 0.142 nm in graphene). As interesting features are observed in the electronic structure of zigzag BSiNRs, and also previous studies [17, 20] pointed out that the buckling form of silicene is more stable than the flat one, hereafter we focus mainly on zigzag BSiNRs and compare them with zigzag GNRs as candidates represent the flat structure.

In this section, the individual effects of a vertical and a transverse electric field on the energy bands of zigzag BSiNRs are considered.

3.2.1. Impact of a vertical electric field

In figures 3(a)–(c), the energy bands of a zigzag BSiNR of width M = 12 under the impact of the vertical electric field for different strengths are displayed (red curves). The results for a zigzag GNR counterpart (black curves) are also shown for comparison. As can be observed, the vertical electric fields do not lead to an opening of a bandgap in the GNR, however, it separates the flat bands of the BSiNR and enlarges the gap of this structure. The enlargement of the bandgap in the BSiNR increases with increasing magnitude of V_t , i.e., with the applied voltage V_t equal to 0.1 V, 0.3 V, and 0.5 V, the obtained bandgaps are equal to 92.2 meV, 292.2 meV, and 492.2 meV, respectively.

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The different phenomena in the electronic structure of the GNR and the BSiNR can be understood based on the description of the TB model presented above. In GNRs, all the carbon atoms lie on the same plane, whereas in BSiNRs, silicon atoms belong to two distinct planes due to the buckling feature of silicene in which the first plane consists of sp²- hybridised atoms, and the other plane contains hybridised ones. When a vertical electric field is applied from top to bottom (as illustrated in figure 1(b)), all carbon atoms in GNRs get the same potential from the vertical electric field, and therefore it does not lead to any difference between atoms in the system. As a result, the energy bands of GNRs are not changed in the presence of a vertical electric field. Meanwhile, for BSiNRs, atoms in the upper and lower planes obtain different potentials ${\rm{\Delta }}-e{V}_{t}/2$ and $-{\rm{\Delta }}+e{V}_{t}/2,$ respectively. This causes an increase in the energy difference between the nearest neighbouring atoms, and enlarges the bandgap in this structure. It is also worth noting that the vertical electric field opens a gap in the considered zigzag BSiNR but it does not change the form of the energy bands as seen in figures 3(a)–(c).

In figure 3(d) the bandgap in the two structures is plotted as a function of the potential of the vertical electric field. The black curve clearly indicates that the bandgap of the GNR remains to be zero regardless of the strength and the direction of the vertical field. On the other hand, the red curve reveals that the gap of the BSiNR increases linearly with increasing magnitude of the vertical electric field. By noting the fact that the intrinsic bandgap is almost equal to $2{\rm{\Delta }}$ that is the energy difference between the energies of the two distinct hybridised Si atoms, thus the dependence of the bandgap on the applied vertical field is expected to follow the same rule, i.e., ${E}_{g}=\left|\left({\rm{\Delta }}-e{V}_{t}/2\right)-\left(-{\rm{\Delta }}+e{V}_{t}/2\right)\right|=\left|2{\rm{\Delta }}-e{V}_{t}\right|$ and this is in agreement with the linear behaviour seen in figure 3(d).

In conclusion, BSiNRs have been seen to be more responsive to a vertical electric field compared to GNRs thanks to the buckling feature which gains the energy difference between the different hybridised atoms. It is also worth mentioning that buckling or out-of-plane deformed graphene ribbons with zigzag edges also do not induce a bandgap under the impact of a vertical electric field due to the strong localisation of edge states at the Fermi level [47]. Thus, the obtained results for zigzag BSiNRs reveal a high potential of BSiNRs for electronic devices where the electronic performance can be flexibly controlled by an external vertical electric field.

3.2.2. Impact of a transverse electric field

To understand the role of the direction of the electric field, we continue to examine the change of the electronic structure in the presence of a transverse electric field.

Figures 4(a)–(c) show the energy bands of the BSiNRs (red line) and the GNRs (black line) for different V_s . It is clear that the transverse electric field has strong impacts on both structures and it is different from the case of the perpendicular electric field that has been considered above. The results demonstrate that the size of the gap increases remarkably in the two structures when increasing the magnitude of V_s as 0.1 V, 0.3 V, and 0.5 V, respectively.

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The variation of the bandgap is directly associated with the change of the originally flat bands (edge states). As can be observed, unlike the case of the vertical electric field, the transverse field has a visible impact on the bands of the edge states when the field strength is significant enough as seen in figures 4(b) and (c). The effect is more pronounced at the two sides of the edge state bands near the points ${k}_{x}{a}_{x}\approx \pm 2\pi /3$ and leads to an in-direct behaviour of the bandgap.

The strong impact of the transverse electric field on the edge states could be understood from the fact that this type of field induces a charge redistribution in the layer in which more electrons move closer to the positive voltage gate while holes move towards the negative ones. As a result, the highest charge density is distributed near the ribbon edges and it becomes smaller when it goes to the centre of the ribbon. Such results are consistent with previous studies [43, 44]. As edge states in zigzag structures localised at the edges, they are strongly affected by the charge redistribution caused by the external field leading to the change in the shape of the band of the edge states.

In further analysis, we considered the variation of the bandgap as a function of the applied gate voltage V_s . The results are shown in figure 4(d) for both the GNR and the BSiNR. When a weak transverse field with V_s below 0.1 V is applied, the gap in both structures is almost the same and increases linearly with increasing V_t . However, above V_t  = 0.1 V, the change of the bandgap is non-linear and there is a discrepancy between the two structures in which the electric field induces a larger gap in the GNR compared to that in the BSiNR. In the range of V_t from −1.8 V to 1.8 V, the gap of the GNR reaches a maximum value of 660 meV at V_t  = ±1.6 V and in the BSiNR this value is about 387.6 meV at V_t  = −0.94 V.

It is worth mentioning that the variation of the bandgap in the GNR is symmetrical with respect to the sign of V_t , however, it is slightly asymmetrical in the case of the BSiNR. Such asymmetrical effect stems from the energy difference of the two types of hybridised Si atoms in BSiNRs. Under the impact of the transverse electric field, the energies of Si atoms in the upper and lower planes of the structure are ${\rm{\Delta }}-e.V\left(i\right)$ and $-{\rm{\Delta }}-e.V\left(j\right),$ respectively. And once the electric field reversed its direction, these energies change to ${\rm{\Delta }}+e.V\left(i\right)$ and $-{\rm{\Delta }}+e.V(j),$ respectively and it is clear that they are not symmetrical with the previous energies. However, because ${\rm{\Delta }}=3.9$ meV is small, the asymmetrical effect is therefore not pronounced.

The above analyses have shown the distinct impacts of a vertical and a transverse electric field on the electronic structure of the flat (zigzag GNRs) and buckling (zigzag BSiNRs) structures. Therefore, it is worth investigating the mutual impact of these two fields on the energy bands. As the vertical electric field does not impact the electronic properties of GNRs, the mutual effect in this material is likely identical to the effect of the transverse field that is considered above. We therefore only examine the mutual effect in BSiNRs.

In figure 5 the bandgap of several zigzag BSiNRs of width M = 10, 11, 12 is plotted as a function of V_t (V_s ) for a certain magnitude of V_s (V_t ).

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First, the results in figures 5(a)−(b) reveal that the impact of the vertical field is almost independent of the ribbon width (or the number of zigzag lines M). In the presence of a transverse field with V_s  = 0.3 V (figure 5(b)), it can be seen that the gap is enlarged for all positive V_t , indicating a stronger modulation of the electronic structure when combining the two fields.

On the other hand, the variation of the gap of the considered BSiNRs without and with a transverse electric field depends on the width of the ribbon as can be clearly observed in figures 5(c) and (d). Interestingly, with the combination of the vertical field with the voltage V_t  = 0.3 V, the asymmetry of the bandgap with respect to the sign of V_s becomes more remarkable as seen in figure 5(d). Besides, the maximum of the bandgaps increases from 459.3 meV, 420.5 meV, and 387.6 meV with V_t  = 0 to 649.7 meV, 612.7 meV and 581.6 meV when V_t  = 0.3 V for the ribbons of width M = 10, 11, 12, respectively.

To give an overview on the modulation of the electronic structure under the impacts of the two coexisting fields, the bandgap is plotted as a function of both V_t and V_s and shown in figure 6(a) for M = 12. Similar results were observed for other ribbons of width.

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The result in figure 6(a) shows that the bandgap in the BSiNR is almost equal to zero if the two fields have opposite potentials V_t  = −V_s . However, the gap is enlarged when the pair of the potentials [V_t , V_s ] is either in region 1 or region 2 at the two sides of the diagonal line V_t  = −V_s . Interestingly, the largest gap is obtained when the two fields are applied simultaneously, i.e., E_g  ≈ 1.2 eV at V_t ≈ −1 V and V_s  ≈ −0.5 V

To better understand the mutual effect in the BSiNR, we make a comparison with the result obtained for a zigzag BGNR as such a structure has two layers of atoms in which both vertical and electric fields affect the electronic structure as indicated in a previous study [6]. The result for a zigzag BGNR of width M = 12 is shown in figure 6(b). As can be observed, the bandgap of the considered BGNR is classified into four regions that are defined by the cones V_s  = V_t and V_s  = −V_t . More importantly, although the vertical field induces an effect in the formation of the bandgap in the BGNR, its role is weak as the largest gap is obtained in the regions (1) and (3) and near the points where V_t ≈ 0. The much weaker effect of the vertical field in the BGNR compared to that in the BSiNR could be interpreted by the fact that although the vertical field induces a significant difference in the onsite energies of atoms in the upper and lower layers of the BGNR, due to the natural weakness of Van de Waals interactions between the two graphene layers, it does not lead to a strong charge redistribution between layers. In contrast, atoms in the BSiNR are also classified into two planes, but they belong to the same layer, so the interaction between atoms in the two planes is much stronger and a charge redistribution is more sensitive to an applied vertical field.

Thus, with the combination of two types of fields, the bandgap of zigzag BSiNRs could be tuned more effectively compared to that of zigzag GNRs and BGNRs. In other words, the simultaneous impact of fields on the buckling structure leads to high efficiency in the modulation of the bandgap of the material. Such results suggest that BSiNRs have favourable advantages to be exploited for different electronic applications.

In this section, we consider the impact of external fields on the thermoelectric transport properties of devices made of zigzag BSiNRs. A device (as illustrated in figure 1(b)) of the length L ≈ 40 nm and the width W ≈ 3.73 nm (M = 12) was considered. The electron transmission was computed by the NEGF technique [48] and the electrical conductance ${G}_{e},$ the Seebeck coefficient S, and the electron thermal conductance K_e were calculated based on the Landauer formalism [37, 49]. The results of ${G}_{e},S$ and the power factor $P={G}_{e}{S}^{2}$ at room temperature and under the impact of external fields are shown in figure 7.

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As can be observed in figure 7(a), a transport gap is opened in the electrical conductance when either a vertical or a transverse field is applied. Such gap is even wider when the two fields are employed simultaneously as can be seen from the purple (dash-dot) curve. The widening of the transport gap leads to a better separation of electrons and holes which results in a larger Seebeck coefficient as can be seen in figure 7(b). The maximum value of the intrinsic Seebeck coefficient of the ribbon in the absence of an external field is about 0.066 mW K⁻¹ as shown in the inset in figure 7(b). This value increases to 0.33 mW K⁻¹ and 0.69 mW K⁻¹ when individual fields of the strengths V_s  = 0.5 V and V_t  = 0.5 V are applied and it even reaches 1.05 mV K⁻¹ with the mutual effect of the two fields. It is worth mentioning that the variation of the electrical conductance and the Seebeck coefficient is fully consistent with the change in the electronic structures in the absence or the presence of external fields, i.e., the decrease (increase) of the electrical conductance (the Seebeck coefficient) around the Fermi level is proportional to the enlargement of the bandgap in the presence of an external field that has been observed in figures 3–6. Also, the quantum number of the electrical conductance (in 2e² /h) at each energy level is equal to the number of channels that can be counted by using a line that crosses the energy bands in a single valley (+k_x or −k_x ) at the corresponding energy level (see figures 3, 4).

The common strategies to enhance the thermoelectric performance of a system are to increase the power factor or/and to reduce the thermal conductance [50–52]. It is therefore more relevant to consider the variation of the power factor. It can be seen in figure 7 that the maximum value of the power factor when V_t  = V_s  = 0 is about 0.59 pW K⁻² and it increases to 0.74 pW K⁻², 0.69 pW K⁻² and 0.66 pW K⁻² when the potentials [V_t , V_s ] are equal to [0.5 V, 0], [0, 0.5 V], and [0.5 V, 0.5 V], respectively. Thus, although the mutual effect has the most impact on the Seebeck coefficient, the vertical field is found to be overall more efficient for enhancing the power factor.

To completely evaluate the thermoelectric performance of the system that also includes the contribution of the phonon part, we calculated the figure of merit ZT by using equation (6) and the value of the phonon conductance was referred from molecular dynamic simulations that were carried out by Pan et al [39]. As the phonon conductance increases almost linearly with increasing ribbon width, we extrapolated K_p at the width M = 12 and obtained the value of 0.1788 nW K⁻¹. It is worth mentioning that the phonon conductance does not change when different fields are applied. Figure 8(a) display ZT for all the considered pairs of the potentials. Due to the small Seebeck coefficient, $Z{T}_{\max }$ of the ribbon in the absence of fields is only about 0.15 and this result is in agreement with that obtained in [39] by density functional theory calculations and molecular dynamic simulations. Interestingly, ZT increases remarkably with the presence of the fields, i.e., $Z{T}_{\max }$ equals 1.02, 0.89 and 0.90 when [V_t , V_s ] are equal to [0.5 V, 0], [0, 0.5 V], and [0.5 V, 0.5 V], respectively. Such enhancement of ZT is about 6–7 times higher in the system without fields. To better understand the strong enhancement of ZT even though the power factor just increases not more than 1.25 times, we examined the electron thermal conductance ${K}_{e}$ that corresponds to the $Z{T}_{\max }$ and the results are shown in figure 8(b). As can be observed, ${K}_{e}$ reduces about an order of magnitude when the fields are applied. The strong degradation of ${K}_{e}$ near the Fermi level is due to the widening of the bandgap that decreases the mobility of electrons. Thus, the enhancement of ZT mainly originates from the reduction of the electron thermal conductance.

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